Emergency Border Security Supplemental Appropriations Act, 2010 - Appropriates additional amounts to remain available until September 30, 2012, for: (1) U.S. Customs and Border Protection use along the Southwest border; (2) border security fencing, infrastructure, and technology along the Southwest border; (3) acquisition and deployment of unmanned aircraft systems; (4) construction of up to three Border Patrol forward operating bases along the Southwest border; (5) U.S. Immigration and Customs Enforcement law enforcement activities along the Southwest border; (6) Federal Emergency Management Agency's (FEMA) Operation Stonegarden (enhancement of law enforcement operational readiness along the U.S. borders); and (7) Department of Justice (DOJ) law enforcement activities related to the Southwest border. Rescinds from unobligated balances certain funds for: (1) U.S. Customs and Border Protection, border security fencing, infrastructure, and technology; (2) Transportation Security Administration (TSA), aviation security; (3) FEMA, administrative and regional operations; and (4) Department of Commerce, Bureau of the Census, periodic censuses and programs. Directs the Department of Defense (DOD) to pay in FY2010-FY2011 the full costs associated with deployment of the National Guard along the Southwest border.